by Hector Chahuara, Anna Krokhine and Elshadai Tegegn<\/p>\n\n\n\n
Controlling caustics is a difficult task as any change to specular surface can have large effects on the caustic image. In this post, we address a building block of the optimization framework for computing the shape of refractive objects presented in Schwartzburg et al (2014) and propose to improve a building block on its formulation that performs a mesh reconstruction. The proposed improvement was tested on flatland meshes and performs reasonably well in the presence of noise.<\/p>\n\n\n\n
Regularization is a technique often needed in optimization for enforcing certain characteristics such as sparsity or stability in the solution. In this particular case, Schwartzburg et al (2014) apply a generalized Tikhonov regularization to achieve a stable solution. Given the optimization-based formulation of Schwartzburg et al (2014), it is possible to isolate this problem and to see that this is in fact mesh denoising. In addition, it is important to mention that Tikhonov regularization is usually outperformed by other methods, among which total variation (TV) denoising distinguishes itself by its reasonable computational cost and good results.<\/p>\n\n\n\n
In the following, we apply the framework described in Zhang et al (2015) that applies TV to the noisy normalized normals N0<\/sub> of a mesh, then the optimization problem to solve becomes<\/p>\n\n\n\n minN<\/sub> ||N-N0<\/sub>||2<\/sup> + \u03bb.WVTV(N),<\/p>\n\n\n\n where WVTV is a weighted vectorial TV version, adapted from the one described in Zhang et al (2015) for flatland, defined as<\/p>\n\n\n\n WVTV(N) = \u03a3e<\/sub>\u03c9e<\/sub> (le<\/sub>(Dx<\/sub>N)2<\/sup>+(Dy<\/sub>N)2<\/sup>)0.5<\/sup><\/p>\n\n\n\n le<\/sub> is the length of the edge \u201ce\u201d, and the weights \u03c9e<\/sub> , dependent on the difference of two consecutive normals Ni<\/sub> and Ni+1<\/sub> i.e. normals that correspond to adjacent edges are defined as<\/p>\n\n\n\n \u03c9e<\/sub>=exp(-||Ni<\/sub>-Ni+1<\/sub>||4<\/sup>)<\/p>\n\n\n\n to penalize less sharp features (high difference between consecutive normals) than the smooth ones (similar consecutive normals). It is important to mention that TV denoising is a non-smooth optimization problem, so a solver that rely solely on gradient or Hessian information could not reach the optimum. By using the iteratively reweighted least squares (IRLS) Wolke et al (1988), a well-known method for optimization, it is straightforward to build an algorithmto solve this problem .<\/p>\n\n\n\n The described approach was implemented in MATLAB R2021b. Visual results can be observed in the following animations that show the denoising process of two figures corrupted by noise: a square and a circle.<\/p>\n\n\n\nResults<\/h2>\n\n\n\n